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The Threshold Probability for Long Cycles

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  08 September 2016

ROMAN GLEBOV ,
HUMBERTO NAVES  and
BENNY SUDAKOV
ROMAN GLEBOV
Affiliation:
School of Computer Science and Engineering, Hebrew University, Jerusalem 9190401, Israel (e-mail: roman.l.glebov@gmail.com)
HUMBERTO NAVES
Affiliation:
Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA (e-mail: hnaves@ima.umn.edu)
BENNY SUDAKOV
Affiliation:
Department of Mathematics, ETH, 8092 Zurich, Switzerland (e-mail: benjamin.sudakov@math.ethz.ch)
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Abstract

For a given graph G of minimum degree at least k, let Gp denote the random spanning subgraph of G obtained by retaining each edge independently with probability p = p(k). We prove that if p ⩾ (logk + loglogk + ωk(1))/k, where ωk(1) is any function tending to infinity with k, then Gp asymptotically almost surely contains a cycle of length at least k + 1. When we take G to be the complete graph on k + 1 vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C38: Paths and cycles 05C85: Graph algorithms 05D40: Probabilistic methods 05C05: Trees
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 2 , March 2017 , pp. 208 - 247
Copyright
Copyright © Cambridge University Press 2016 

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